How is the Apple Mini DisplayPort to Dual-Link DVI adapter different from the Apple Mini DisplayPort to DVI adapter?

The Apple Mini DisplayPort to Dual-Link DVI adapter works best with displays that operate at dual-link resolutions, which are resolutions above 1920 x 1200. For displays that do not support dual-link DVI resolutions (displays that operate at resolutions of 1920 x 1200 or lower) you should use the Apple Mini DisplayPort to DVI adapter instead.

Also, the Wikipedia article on DVI connectors gives you images of the different connectors so you can see what your monitor has.